Mathematics for the interested outsider

Outer billiards has an unbounded orbit

This just popped up on the arXiv, so I thought I should mention it: Richard Schwartz has a paper up showing that there is a shape for an “outer billiards” table and a starting point whose path gets as far away from the shape as you want. Even better, it’s one of the shapes from the Penrose tiling. Curiouser and curiouser. The first section or so of the paper are very readable, and gives a much better explanation (with pictures!) of outer billiards than I could manage. The proof itself is heavily aided by computer calculations, but seems to be tightly reasoned apart from needing help to handle a lot of cases.

I’m not quite sure how billiards and outer billiards are related. My intuition is that there’s some sort of duality going on, which would exchange lengths of segments in outer billiards with angles in billiards. On the other hand, if there were such a straightforward translation, couldn’t the enormous machinery of billiards have been brought to bear on this problem before now? Do any billiard theorists in the audience know anything about outer billiards?

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[…] The last time I talked about billiards I was linking to Rich Schwartz’ paper on “outer billiards”. I noted that it seemed to me there should be some sort of “duality” between outer and inner billiards, turning problems in one into problems in the other. I think I’ve figured it out. I haven’t checked through all the details, but it looks good enough to satisfy my curiosity. If I were going to write a paper and use this fact, of course, I’d rake it over the coals. […]

I (the author of the paper on unbounded orbits for outer billiards)
don’t really think that there is such a strong connection between
outer and ordinary billiards. One strong difference between these
two kinds of systems is that ordinary billiards is natural with respect
to the similarity group of the plane whereas outer billiards is natural
with respect to the affine group. Also, various questions which are
extremely difficult for ordinary billiards – e.g. the existence of
periodic orbits – are much easier for outer billiards. My latest (not
yet published) work on outer billiards suggests that polygonal
outer billiards is in some sense governed by polytope exchange maps
on high dimensional tori, whereas nothing even remotely like this
is known (or should be true) for ordinary billiards.

I think that “outer billiards” is miscalled “billiards”. It seems to me more
reminiscent of “bad golf”, where you are trying to putt the ball into
the hole, but keep missing and having to chase the ball around the
green.

You might, then, be able to say something about my later idea, that outer billiards and inner billiards are projective duals of each other. As I say there, I haven’t sat down to crank out all the coordinate transformations, but it seems like it should work. Of course, the transformations may be too complicated to use in practice to transfer statements about one system to the other.

I do like the “bad golf” label, though. Maybe a collaboration with LeslieNielsen?

(Hi John) I would be interested to see this sort of coordinate transform that
you mention, though I doubt it will work in anything but in special cases.
For instance, I’m sure it will work for ellipses. People who have mentioned to
me connections between inner and outer billiards frequently cite the ellipse
as a classical example where things work out well and then cannot suggest
any other kind of example. Well, perhaps something works out in the smooth case… I’ve been very focused on polygonal billiards.

Consider the case of the triangle. Presumably you would like to say that
there is an analogy between outer billiards on triangles and inner billiards
on triangles, but nothing like this could possibly work. Outer billiards is
the same for any triangle, up to affine equivalence of the orbits, whereas
inner billiards on triangles is a very deep and still poorly understood
subject. If you could really find a coordinate transformation that works
for the triangles, you would have an easy solution to some problems
that are over 200 years old, such as the triangular billiards conjecture.
This is the conjecture that inner billiards always has a periodic orbit in
any triangle. If you have in mind an “information preserving” transformation
between triangular inner billiards and some other kind of outer billiards
then this would be an amazing result and I would kill to get my hands
on it.

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